Entanglement in Chern Simons theory and edge states in Haldane spin chains

Abstract

In the first part of this thesis, we review Chern Simons theory and study its topological entanglement entropy using the replica trick. We review the quantisation of Chern Simons theory and see how this seemingly trivial theory has a rich structure and a deep connection with chiral conformal field theory which lets us calculate partition function on one manifold from one on another manifold. We see why this calculation is difficult to do for Chern Simons theory coupled with matter. In the second part, we look at 1 D antiferromagnetic spin chains and their edge states and see how their low energy physics can be mapped to the Op3q quantum rotor model. We look at an exactly solvable model, the AKLT model of spin 1 particles and discuss a pictorial way to see how edge states emerge. We exactly diagonalize S “ 1 Heisenberg spin chain to look at the Haldane gap and see how the parity of length of the spin chain decides the ground state. We support these numerical results with an analytical study of the effective Hamiltonian between the edge spins and by deriving an effective Hamiltonian between them.